Method in connection with sensorless induction motors

ABSTRACT

A method for the stabilization of full-order flux observers for speed-sensorless induction motors in the regenerative mode. The method comprises determining the current vector of the induction motor, determining the stator voltage vector of the induction motor, forming a full-order flux observer having a system matrix (A) and a gain matrix (L), the state-variable observer being augmented with a speed adaptation loop, and producing an estimated rotor flux linkage vector and an estimated stator current vector, determining an estimation error of the stator current vector, defining a correction angle, and forming a speed adapt-tion law based on the cross product of the estimation error of the stator current vector and the estimated rotor flux linkage vector, where the correction angle is used to turn the rotor flux linkage vector or the estimation error of the stator current vector in order to keep the observer stable.

FIELD OF THE INVENTION

The present invention relates to use a of full-order flux observers, andparticularly to stabilization of the full-order flux observers forspeed-sensorless induction motors in the regenerative mode.

BACKGROUND OF THE INVENTION

Speed-sensorless induction motor drives have developed significantlyduring the past few years. Speed-adaptive full-order flux observers arepromising flux estimators for induction motor drives. The speed-adaptiveobserver consists of a state-variable observer augmented with aspeed-adaptation loop. The observer gain and the speed-adaptation lawdetermine the observer dynamics.

The conventional speed-adaptation law was originally derived using theLyapunov stability theorem or the Popov hyperstability theorem. However,the stability of the adaptation law is not guaranteed since,controversial assumptions regarding nonmeasurable states have been usedin and the positive-realness condition is not satisfied in. An unstableregion encountered in the regenerating mode at low speeds is well known.The regenerating-mode low-speed operation is problematic also for theestimators based on the voltage model as shown in.

In the case of the speed-adaptive full-order flux observer, the size ofthe unstable region could be reduced or, in theory, even removed bychoosing the observer gain suitably. However, based on the simulationscarried out, the methods and are sensitive to very small errors in themotor parameters. Furthermore, a seamless transition from theregenerating-mode low-speed operation to higher-speed operation ormotoring-mode operation may be problematic.

Another approach to remedy the instability is to modify thespeed-adaptation law. This approach seems to be almost unexplored. Inchanging the direction of the error projection of the adaptation law wasdiscussed (but not Studied) for a filtered back-emf-based observer. Inthe rotor flux estimate included in the adaptation law was replaced withthe stator flux estimate, but this does not remove the unstable region.

BRIEF DESCRIPTION OF THE INVENTION

An object of the present invention is to provide a method so as to solvethe above problem. The object of the invention is achieved by a method,which is characterized by what is stated in the independent claim. Thepreferred embodiments of the invention are disclosed in the dependentclaims.

The method of the invention is based on a modified speed-adaptation lawwhere the direction of the error projection is changed in theregenerating-mode low-speed operation. Thus instead of using only thecurrent estimation error perpendicular to the estimated flux, theparallel component is also exploited in the regenerating mode.

An advantage of the method of the invention is that the control ofsensorless induction motor will be stabile in all operating pointsincluding low-speed regeneration. The control of an induction motorbased on the method of the invention is fast to implement and reliable.

BRIEF DESCRIPTION OF THE DRAWINGS

In the following the invention will be described in greater detail bymeans of preferred embodiments with reference to the attached drawings,in which

FIGS. 1 and 2 illustrate loci of the current estimation error;

FIG. 3 a illustrates the linearized model of the observer using theprior art speed-adaptation law;

FIG. 3 b illustrates the linearized model of the observer using thespeed adaptation law according to the invention;

FIG. 4 a shows a part of the root loci of the prior art speed adaptationlaw;

FIG. 4 b shows a part of the root loci of the inventive speed adaptationlaw;

FIG. 5 illustrates the experimental setup;

FIG. 6 illustrates a rotor-flux-oriented controller;

FIGS. 7 a-b and 8 illustrate experimental results in the regeneratingmode;

FIG. 9 a illustrates simulation results; and

FIG. 9 b illustrates experimental results.

DETAILED DESCRIPTION OF THE INVENTION

In the description the induction motor model and the speed-adaptive fluxobserver are first defined. Then, steady-state analysis is used toclarify the problem underlying the invention and its solution accordingto the invention. The stability is also studied by using root loci ofthe linearized system. Finally, after describing a control system basedon the rotor flux orientation, simulation and experimental results arepresented.

Induction Motor Model

The parameters of the dynamic Γ-equivalent circuit of an induction motorare the stator resistance R_(s), the rotor resistance R_(R), the statortransient inductance L′_(s), and the magnetizing inductance L_(M). Theangular speed of the rotor is denoted by ω_(m), the angular speed of thereference frame ω_(k), the stator current space vector is i _(s), andthe stator voltage u _(s). When the stator flux ψ _(s) and the rotorflux ψ _(R) are chosen as state variables, the state-spacerepresentation of the induction motor becomes $\begin{matrix}{\underset{\_}{\overset{.}{X}} = {{\underset{\underset{\underset{\_}{A}\quad}{︸}}{\begin{bmatrix}{{- \frac{1}{\tau_{s}^{\prime}}} - {j\quad\omega_{k}}} & \frac{1}{\tau_{s}^{\prime}} \\\frac{1 - \sigma}{\tau_{r}^{\prime}} & {{- \frac{1}{\tau_{r}^{\prime}}} - {j\quad\left( {\omega_{k} - \omega_{m}} \right)}}\end{bmatrix}}\underset{\_}{X}} + {\underset{\underset{B}{︸}}{\begin{bmatrix}1 \\0\end{bmatrix}}{\underset{\_}{u}}_{s}}}} & \left( {1a} \right) \\{{\underset{\_}{i}}_{s} = {\underset{\underset{C}{︸}}{\begin{bmatrix}\frac{1}{L_{s}^{\prime}} & {- \frac{1}{L_{s}^{\prime}}}\end{bmatrix}}\underset{\_}{X}}} & \left( {1b} \right)\end{matrix}$where the state vector is X=[ψ _(s) ψ _(R)]^(T), and the parametersexpressed in terms of the Γ-equivalent circuit parameters areσ=L′_(s)/(L_(M)+L′_(s)), τ′_(s)=L′_(s)/R_(s), and τ′_(r)=σL_(M)/R_(R).The electromagnetic torque is $\begin{matrix}{T_{e} = {{\frac{3}{2}p\quad{Im}\left\{ {{\underset{\_}{i}}_{s}{\underset{\_}{\psi}}_{R}^{*}} \right\}} = {\frac{3}{2}p\frac{1}{L_{s}^{\prime}}{Im}\left\{ {{\underset{\_}{\psi}}_{s}{\underset{\_}{\psi}}_{R}^{*}} \right\}}}} & (2)\end{matrix}$where p is the number of pole pairs and the complex conjugates aremarked by the symbol *. In the specification, the parameters of a 2.2-kWfour-pole induction motor given in Table I are used. It should also beunderstood that these parameters are used only for explaining theinvention.

The method according to the invention comprises determining the currentvector of the induction motor and determining the stator voltage vectorof the induction motor. The current vector is obtained, for example, bymeasuring the currents. In a three-phase system it is usually necessaryto measure only two currents.

The voltage vector is obtained, for example, by measuring the voltage inthe apparatus feeding the motor. The apparatus is usually a frequencyconverter with a direct voltage intermediate circuit. By measuring thisvoltage and combining it with state information of the output switches,the output voltage vector is achieved.

Speed-Adaptive Full-Order Flux Observer

Conventionally, the stator current and the rotor flux are used as statevariables in full-order flux observers. However, choosing the stator androtor fluxes as state variables is preferred since this allows theobserver to be used with stator-flux-oriented control or direct torquecontrol [8] as well as with rotor-flux-oriented control. Consequently,the full-order flux observer is defined by $\begin{matrix}{\overset{\overset{.}{\hat{}}}{\underset{\_}{X}} = {\underset{\_}{\hat{A}\hat{X}} + {B\quad{\underset{\_}{u}}_{s}} + {\underset{\_}{L}\left( {{\underset{\_}{i}}_{s} - {\hat{\underset{\_}{i}}}_{s}} \right)}}} & \left( {3a} \right)\end{matrix}$î _(s) =C{circumflex over (X)}   (3b)where the observer state vector is X=[{circumflex over (ψ)} _(s){circumflex over (ψ)} _(R)]^(T), and the system matrix and the observergain are $\begin{matrix}{{\underset{\_}{\hat{A}} = \begin{bmatrix}{{- \frac{1}{\tau_{s}^{\prime}}} - {j\quad\omega_{k}}} & \frac{1}{\tau_{s}^{\prime}} \\\frac{1 - \sigma}{\tau_{r}^{\prime}} & {{- \frac{1}{\tau_{r}^{\prime}}} - {j\quad\left( {\omega_{k} - {\hat{\omega}}_{m}} \right)}}\end{bmatrix}},{\underset{\_}{L} = \begin{bmatrix}{\underset{\_}{l}}_{s} \\{\underset{\_}{l}}_{r}\end{bmatrix}}} & \left( {3c} \right)\end{matrix}$respectively, where the estimates are marked by the symbol ^.Observer Gain

The simple observer gainl _(s)=λ[1+jsgn({circumflex over (ω)}_(m))], l_(r)=λ[−1+jsgn({circumflex over (ω)}_(m))]  (4a)where $\lambda = \left\{ \begin{matrix}{\lambda^{\prime}\frac{{\hat{\omega}}_{m}}{\omega_{\lambda}}} & {{{if}\quad{{\hat{\omega}}_{m}}} < \omega_{\lambda}} \\\lambda^{\prime} & {{{if}\quad{{\hat{\omega}}_{m}}} \geq \omega_{\lambda}}\end{matrix} \right.$gives satisfactory behavior from zero speed to very high speeds.Parameters λ′ and ω₈₀ are positive constants. The parameter can beconsidered as an impedance, which may be helpful when choosing λ′ fordifferent motor sizes. In the specification, the observer gain isdetermined by λ′=1 p.u.Speed-Adaptation LawsPrior Art

The conventional speed-adaptation law is{circumflex over (ω)}_(m)=−γ_(p) Im{( i _(s) −î _(s)){circumflex over(ψ)}*_(R)}−γ_(i) ∫Im{( i _(s) −î _(s)){circumflex over (ψ)}*_(R)}dt  (5)where γ_(p) and γ_(i) are the adaptation gains. Only the currentestimation error perpendicular to the estimated rotor flux is used toestimate the speed. The adaptation law works well except in theregenerating mode at low speeds. The gains γ_(p)=10 (Nm·s)⁻¹ andγ_(i)=10000 (Nm·s²)⁻¹ are used in this specification.According to the Present Invention

The speed-adaptation law according to the present invention is{circumflex over (ω)}_(m)=−γ_(p) Im{( i _(s) −î _(s)){circumflex over(ψ)}*_(R) e ^(−jφ)}−γ_(i) ∫Im{( i _(s) −î _(s)){circumflex over(ψ)}*_(R) e ^(−jφ) }dt  (6)where the angle φ changes the direction of the error projection. Inother words, the component of the current estimation error parallel tothe estimated rotor flux is also exploited when φ≠0. The change in thedirection of the error projection is needed to stabilize theregenerating-mode operation at low speeds. Equation (6) is simple tocalculate since Im{a b*} can be interpreted as the cross product of thevectors. In the case of (6) the cross product is calculated betweenstator current estimation error and estimated rotor flux.

In the speed adaptation the estimated rotor flux linkage is used. Themethod is also applicable for estimating stator flux linkage. Thisallows the method to be used in a wide variety of vector controlmethods.

Steady-State Analysis

Based on (1) and (3), the estimation error e=X−{circumflex over (X)} ofthe state vector and the stator current error are $\begin{matrix}{\underset{\_}{\overset{.}{e}} = {{\left( {\underset{\_}{A} - {\underset{\_}{L}\quad C}} \right)\underset{\_}{e}} + {\begin{bmatrix}0 \\{j\quad{\underset{\_}{\hat{\psi}}}_{R}}\end{bmatrix}\left( {\omega_{m} - {\hat{\omega}}_{m}} \right)}}} & \left( {7a} \right) \\{{{\underset{\_}{i}}_{s} - {\underset{\_}{\hat{i}}}_{s}} = {C\quad\underset{\_}{e}}} & \left( {7b} \right)\end{matrix}$

In the following, the estimation error e is considered in the steadystate and the estimated rotor flux reference frame is used, i.e., ė=0,ω_(k)=ω_(s) (where ω_(s) is the angular stator frequency), and{circumflex over (ψ)} _(R)={circumflex over (ψ)}_(R)+j0. For a givenerror ω_(m)−{circumflex over (ω)}_(m), and an operating point determinedby the angular stator frequency ω_(s), the angular slip frequencyω_(r)=ω_(s)−ω_(m), and the rotor flux estimate {circumflex over(ψ)}_(R), a steady-state solution for (7) can be easily found.

Stable Region

FIG. 1 depicts the loci of current estimation error for two differentspeed estimation errors when the angular slip frequency ω_(r) variesfrom the negative rated slip to the positive rated slip. The angularstator frequency is ω_(s)=0.1 p.u. and the base value of the angularfrequency is 2π50 s⁻¹. It can be seen that the larger the speed error,the larger the current estimation error.

FIG. 1 shows the loci of the current estimation error when the angularslip frequency ω_(r) varies from the negative rated slip to the positiverated slip (the rated slip being 0.05 p.u.). The angular statorfrequency is ω_(s)=0.1 p.u. and two different speed estimation errors(0.002 p.u. and 0.004 p.u.) are shown. The estimated rotor fluxreference frame is used in FIG. 1.

In FIG. 1, ω_(s)>0 and {circumflex over (ω)}_(m)>ω_(m). If ω_(s)<0, theloci lie in the right half-plane. If {circumflex over (ω)}_(m)<ω_(m),the loci are located in the lower half-plane.

In the estimated rotor flux reference frame, the prior art adaptationlaw (5) reduces to{circumflex over (ω)}_(m)=−γ_(p)(i _(sq) −i _(sq)){circumflex over(ψ)}_(R)−γ_(i)∫(i _(sq) −î _(sq)){circumflex over (ψ)}_(R) dt  (8)

The speed estimate thus depends on the error i_(sq)−î_(sq). If{circumflex over (ω)}_(m)>ω_(m), the condition i_(sq)−î_(sq)>0 shouldhold in order the speed estimate to converge. In FIG. 1, this conditionholds for all slip frequencies including the regenerating-mode operation(where ω_(s)ω_(r)<0).

Unstable Region

FIG. 2 shows loci of the current estimation error for a lower angularstator frequency ω_(s)=0.01 p.u. The locus consisting of the solid curveand the dashed curve shows the current estimation error. The conditioni_(sq)−î_(sq)>0 holds in the motoring-mode operation, but in theregenerating-mode operation at higher slips, it does not hold. Hence,the observer using the prior art adaptation law becomes unstable.

FIG. 2 shows loci of the current estimation error when the angular slipfrequency ω_(r) varies from the negative rated slip to the positiverated slip. The angular stator frequency is ω_(s)=0.01 p.u. and thespeed estimation error is {circumflex over (ω)}_(m)−ω_(m)=0.002 p.u. Thedashed/solid curve shows the locus corresponding to the prior artadaptation law. The locus consisting of the solid curve and thedash-dotted curve corresponds to the adaptation law as used inconnection with the present invention. In FIG. 2 the estimated rotorflux reference frame is used.

Based on FIG. 2, it can be noticed that the regenerating mode can bestabilized by changing the direction of the error projection.Consequently, the adaptation law (6) according to the method of theinvention in the estimated rotor flux reference frame is considered. Thecurrent estimation error is rotated by factor exp(−jφ) in the estimatedflux reference frame. Since the prior art adaptation law works well inthe motoring mode, the angle φ is selected as $\begin{matrix}{\phi = \left\{ \begin{matrix}{\phi_{\max}{{sgn}\left( \omega_{s} \right)}\left( {1 - \frac{\omega_{s}}{\omega_{\phi}}} \right)} & {{{if}\quad{\omega_{s}}} < {\omega_{\phi}\quad{and}\quad\omega_{s}{\hat{\omega}}_{r}} < 0} \\0 & {otherwise}\end{matrix} \right.} & (9)\end{matrix}$

For the given motor, φ_(max)=0.44π (i.e., 80°) and ω_(φ)=0.4 p.u. werechosen. In FIG. 2, the current error locus resulting from (9) consistsof the solid curve and the dash-dotted curve, i.e., the dashed curve wasrotated 78° around the origin in order to obtain the dash-dotted curve.Now, the condition i_(sq)−î_(sq)>0 is valid for all slip frequencies.Actually, the selection (9) stabilizes the whole regenerating region.The parameters φ_(max) and ω_(φ) can be substantially varied withoutlosing the stability.

The adaptation law according to the inventive method is not restrictedto the observer gain (4). Several observer gains were studied using thesteady-state analysis and the linearized model. Even the same values ofφ_(max) and ω_(φ) as with the observer gain (4) can be used in somecases, e.g., when using the observer gain proposed in or the zeroobserver gain.

Linearized Model

The nonlinear and complicated dynamics of the speed-adaptive observercan be studied via linearization. The key factor in the linearization isto use a synchronous reference frame in order to obtain a dc equilibriumpoint. In the following, the dynamics of both the motor and the observerare taken into account. Even though the stator dynamics are included inthe model, the linearized model is independent of the stator voltageand, consequently, of the current controller.

Estimation Error

In the rotor flux reference frame, the linearized model of (7a) becomes[11] $\begin{matrix}{\underset{\_}{\overset{.}{e}} = {{\left( {{\underset{\_}{A}}_{0} - {{\underset{\_}{L}}_{0}C}} \right)\underset{\_}{e}} + {\begin{bmatrix}0 \\{j\quad\psi_{R0}}\end{bmatrix}\left( {\omega_{m} - {\hat{\omega}}_{m}} \right)}}} & \left( {10a} \right)\end{matrix}$

Here, the equilibrium point quantities are marked by the subscript 0,and the system matrix and the observer gain are $\begin{matrix}{{{\underset{\_}{A}}_{0} = \begin{bmatrix}{{- \frac{1}{\tau_{s}^{\prime}}} - {j\quad\omega_{s0}}} & \frac{1}{\tau_{s}^{\prime}} \\\frac{1 - \sigma}{\tau_{r}^{\prime}} & {{- \frac{1}{\tau_{r}^{\prime}}} - {j\quad\omega_{r0}}}\end{bmatrix}},{{\underset{\_}{L}}_{0} = \begin{bmatrix}{\underset{\_}{l}}_{s0} \\{\underset{\_}{l}}_{r0}\end{bmatrix}}} & \left( {10b} \right)\end{matrix}$respectively.

The transfer function from the estimation error of the speedω_(m)−{circumflex over (ω)}_(m) to the estimation error of the current i_(s)−î _(s) is $\begin{matrix}\begin{matrix}{{\underset{\_}{G}(s)} = {{C\left( {{sI} - {\underset{\_}{A}}_{0} + {{\underset{\_}{L}}_{0}C}} \right)}^{- 1}\begin{bmatrix}0 \\{j\quad\psi_{R0}}\end{bmatrix}}} \\{= {{- \frac{{- j}\quad\psi_{R0}}{L_{s}^{\prime}}}\frac{s + {j\quad\omega_{s0}}}{{A(s)} + {j\quad{B(s)}}}}}\end{matrix} & \left( {11a} \right)\end{matrix}$where $I = \begin{bmatrix}1 & 0 \\0 & 1\end{bmatrix}$is the identity matrix. The polynomials in (11a) are defined as$\begin{matrix}{{A(s)} = {s^{2} + {s\left( {\frac{1}{\tau_{s}^{\prime}} + \frac{1}{\tau_{r}^{\prime}} + \frac{l_{{sd}\quad 0} - l_{r\quad d\quad 0}}{L_{s}^{\prime}}} \right)} - {\omega_{s0}\omega_{r0}} + \frac{\sigma}{\tau_{s}^{\prime}\tau_{r}^{\prime}} + \frac{{\omega_{s\quad 0}l_{{rq}\quad 0}} - {\omega_{r\quad 0}l_{{sq}\quad 0}}}{L_{s}^{\prime}} + \frac{\sigma_{{lsd}\quad 0}}{\tau_{r}^{\prime}L_{s}^{\prime}}}} & \left( {11b} \right) \\{{B(s)} = {{s\left( {\omega_{s0} + \omega_{r0} + \frac{l_{{sq}\quad 0} - l_{r\quad q\quad 0}}{L_{s}^{\prime}}} \right)} + \frac{{\omega_{s0}\tau_{s}^{\prime}} + {\omega_{r0}\tau_{r}^{\prime}}}{\tau_{s}^{\prime}\tau_{r}^{\prime}} + \frac{{\omega_{r\quad 0}l_{{sd}\quad 0}} - {\omega_{s\quad 0}l_{{rd}\quad 0}}}{L_{s}^{\prime}} + \frac{\sigma\quad l_{{sq}\quad 0}}{\tau_{r}^{\prime}L_{s}^{\prime}}}} & \left( {11c} \right)\end{matrix}$where the entries of the observer gain are divided into real andimaginary components: l _(s0)=l_(sd0)+jl_(sq0) and l_(r0)=l_(rd0)+jl_(rq0). Since the observer gain is allowed to be afunction of the estimated rotor speed, the subscript 0 is used. It is tobe noted that G(s) is independent of the speed-adaptation law.Closed-Loop SystemPrior Art Adaptation Law

Based on the conventional adaptation law (8), the linearized transferfunction from the current error i _(sq)−î _(sq) to the speed estimate{circumflex over (ω)}_(m) is $\begin{matrix}{{K(s)} = {{- \left( {\gamma_{p0} + \frac{\gamma_{i0}}{s}} \right)}\psi_{R0}}} & (12)\end{matrix}$where the gains can be functions of the speed estimate. Only theimaginary component i _(sq)−î _(sq) of the estimation error of thecurrent is of interest. Thus only the imaginary component of (11a) isused, $\begin{matrix}{{G_{q}(s)} = {{{Im}\left\{ {\underset{\_}{G}(s)} \right\}} = {{- \frac{\psi_{R0}}{L_{s}^{\prime}}}\frac{{{sA}(s)} + {\omega_{s0}{B(s)}}}{{A^{2}(s)} + {B^{2}(s)}}}}} & (13)\end{matrix}$

Using (12) and (13), the closed-loop system shown in FIG. 3(a) isformed. The closed-loop transfer function corresponding to any operatingpoint can be easily calculated using suitable computer software (e.g.,MATLAB Control System Toolbox).

FIG. 4(a) shows the root loci of the linearized closed-loop systemcorresponding to the regenerating-mode operation. The slip frequency israted and negative. Only the dominant poles are shown. As assumed, thesystem is unstable at low stator frequencies (a real pole is located inthe right half-plane).

Adaptation Law According to the Invention

In the estimated rotor flux reference frame, the inventive adaptationlaw (6) becomes{circumflex over (ω)}_(m)=−γ_(p)└(i _(sq) −î _(sq))cos(φ)−(i _(sd) −î_(sd))sin(φ)┘{circumflex over (ψ)}_(R)−γ_(i)∫[(i _(sq) −î_(sq))cos(φ)−(i _(sd) −î _(sd))sin(φ)]{circumflex over (ψ)}_(R) dt  (14)

The linearized system is shown in FIG. 3(b), where the transfer functionfrom the estimation error of the speed, ω_(m)−{circumflex over (ω)}_(m)to the estimation error of the current i _(sd)−î _(sd) is$\begin{matrix}{{G_{d}(s)} = {{{Re}\left\{ {\underset{\_}{G}(s)} \right\}} = {{- \frac{\psi_{R0}}{L_{s}^{\prime}}}\frac{{{sB}(s)} - {\omega_{s0}{A(s)}}}{{A^{2}(s)} + {B^{2}(s)}}}}} & (15)\end{matrix}$

FIG. 4(b) shows the root loci of the linearized closed-loop systemcorresponding to the regenerating-mode operation. In this case, thesystem is stable also at low stator frequencies (marginally stable whenthe stator frequency is zero).

FIGS. 4(a) and 4(b) show part of the root loci showing the dominantpoles in the regenerating mode. The slip frequency is rated andnegative. Due to symmetry, only the upper half-plane is shown in theFIGS. 4(a) and 4(b).

Control System

The regenerating-mode low-speed operation of the speed-adaptive observerwas investigated by means of simulations and experiments. TheMATLAB/Simulink environment was used for the simulations. Theexperimental setup is shown in FIG. 5. The 2.2-kW four-pole inductionmotor (Table I) was fed by a frequency converter controlled by a dSpaceDS1103 PPC/DSP board. The PM servo motor was used as the loadingmachine.

The control system was based on the rotor flux orientation. Thesimplified overall block diagram of the system is shown in FIG. 6, wherethe electrical variables on the left-hand side of the coordinatetransformations are in the estimated flux reference frame and thevariables on the right-hand side are in the stator reference frame. Thedigital implementation of the observer proposed in [10] was used. Theflux reference was 0.9 Wb.

A PI-type synchronous-frame current controller was used. The bandwidthof the current controller was 8 p.u. The speed estimate was filteredusing a first-order low-pass filter having the bandwidth of 0.8 p.u, andthe speed controller was a conventional PI-controller having thebandwidth of 0.16 p.u. The flux controller was a PI-type controllerhaving the bandwidth of 0.016 p.u.

The sampling was synchronized to the modulation and both the switchingfrequency and the sampling frequency were 5 kHz. The dc-link voltage wasmeasured, and the reference voltage obtained from the current controllerwas used for the flux observer. A simple current feedforwardcompensation for dead times and power device voltage drops was applied.

It is also understood that the experimental setup is illustrated hereonly for an example. The control system using the method of theinvention can be any known system and is not limited to the mentionedrotor-flux-oriented system.

Results

The base values used in the following figures are: current √{square rootover (2)}·5.0 A and flux 1.0 Wb. Experimental results obtained using theprior art adaptation law are shown in FIG. 7(a). The speed reference wasset to 0.08 p.u. and a negative rated-load torque step was applied att=1 s. After applying the negative load, the drive should operate in theregenerating mode. However, the system becomes unstable soon after thetorque step. According to the root loci of FIG. 4(a), the operatingpoint is unstable since the stator frequency is approximately 0.05 p.u.FIG. 7(b) depicts experimental results obtained using the adaptation lawaccording to the invention. As expected based on the root loci of FIG.4(b), the system behaves stably.

The first subplot of FIGS. 7(a) and 7(b) shows the measured speed(solid) and the estimated speed (dotted). The second subplot shows the qcomponent of the stator current in the estimated flux reference frame.The third subplot presents the real and imaginary components of theestimated rotor flux in the stator reference frame.

FIG. 8 shows experiment results obtained using the adaptation lawaccording to the invention. The speed reference was now set to 0.04 p.u.and the negative rated-load torque step was applied at t=5 s. Eventhough the stator frequency is only about 0.0085 p.u., both the flux andspeed are correctly observed. The explanation of curves are as in FIG.7.

Simulation results showing slow speed reversals are shown in FIG. 9(a).The adaptation law according to the invention was used. A rated-loadtorque step was applied at t=1 s. The speed reference was slowly rampedfrom 0.06 p.u. (t=5 s) to −0.06 p.u. (t=20 s) and then back to 0.06 p.u.(t=35 s). The drive operates first in the motoring mode, then in theregenerating mode, and finally again in the motoring mode.

Corresponding experimental results are shown in FIG. 9(b). The noise inthe current and the speed estimate originates mainly from the incompletedead-time compensation. At a given speed, the proportional effect of thedead-time compensation is more significant in the regenerating mode thanin the motoring mode since the amplitude of the stator voltage issmaller. This kind of speed reversals require a very accurate statorresistance estimate since the stator frequency remains in the vicinityof zero for a long time. If desired, the observer could be augmentedwith a stator resistance adaptation scheme, Experimental results in themotoring-mode operation (demonstrating e.g. zero-speed operation) of thesame speed-adaptive observer can be found in. The explanations of thecurves are as in FIG. 7.

It will be obvious to a person skilled in the art that, as technologyadvances, the inventive concept can be implemented in various ways. Theinvention and its embodiments are not limited to the examples describedabove but may vary within the scope of the claims.

TABLE I PARAMETERS OF THE 2.2-KW FOUR-POLE 400-V 50-Hz MOTOR. Statorresistance R_(s) 3.67 Ω Rotor resistance R_(R) 2.10 Ω Magnetizinginductance L_(M) 0.224 H Stator transient inductance L's 0.0209 H Momentof inertia Jtot 0.0155 kgm² Rated speed 1430 rpm Rated current 5.0 ARated torque 14.6 Nm

1. A method for the stabilization of full-order flux observers forspeed-sensorless induction motors in the regenerative mode,characterized by determining a current vector of the induction motor,determining a stator voltage vector of the induction motor, forming afull-order flux observer having a system matrix (A) and gain matrix (L),the state-variable observer being augmented with a speed adaptationloop, and producing an estimated rotor flux linkage vector and anestimated stator current vector, determining an estimation error of thestator current vector, defining a correction angle, and forming a speedadaptation law based on the cross product of the estimation error of thestator current vector and the estimated rotor flux linkage vector, wherethe correction angle is used to turn the rotor flux linkage vector orthe estimation error of the stator current vector in order to keep theobserver stable.
 2. A method according to claim 1, characterized in thatthe method further comprises controlling the speed-sensorless inductionmotor with the information received from the full-order flux observer,the information comprising the stator or rotor flux linkage vector andthe angular speed of the motor.